Problem: $f(x) = \begin{cases} -1 & \text{if } x = -2 \\ -x^{2}+2 & \text{otherwise} \end{cases}$ What is the range of $f(x)$ ?
Solution: First consider the behavior for $x \ne -2$ Consider the range of $-x^{2}$ The range of $x^2$ is $\{\, y \mid y \ge 0 \,\}$ Multiplying by $-1$ flips the range to $\{\, y \mid y \le 0 \,\}$ To get $-x^{2}+2$ , we add $2$ If $x = -2$, then $f(x) = -1$. Since $-1 ≤ 2$, the range is still $\{\, y \mid y ≤ 2 \,\}$.